Problem: All of the 4th grade teachers and students from Oak went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$2.50$ each for students, and the group paid $$47.00$ in total. The next month, the same group visited a science museum where the tickets cost $$16.50$ each for teachers and $$10.00$ each for students, and the group paid $$166.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+2.5y = 47}$ ${16.5x+10y = 166}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-16.5x-7.5y = -141}$ ${16.5x+10y = 166}$ Add the top and bottom equations together. $ 2.5y = 25 $ $ y = \dfrac{25}{2.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {5.5x+2.5y = 47}$ to find $x$ ${5.5x + 2.5}{(10)}{= 47}$ $5.5x+25 = 47$ $5.5x = 22$ $x = \dfrac{22}{5.5}$ ${x = 4}$ You can also plug ${y = 10}$ into $ {16.5x+10y = 166}$ and get the same answer for $x$ ${16.5x + 10}{(10)}{= 166}$ ${x = 4}$ There were $4$ teachers and $10$ students on the field trips.